5 Stars and Stellar Structure 5.1 Phenomenology of stars Almost all the lightwe see in the Universe comes from stars, either directly, or else indirectly from the fact that stars heat the surrounding gas and dust. Almost the only exceptions to this rule are non-thermal radiation (synchrotron radiation, for example) from high energy particles spiraling around in magnetic elds emission from quasars, thought to b e from the gravitational p otential energy released as matter spirals, via an accretion disk, into a black hole. Wehave already seen that stars can b e classi ed, purely empirically,on the basis of their sp ectra: O B A F G K M R N S. This is prettymucha classi cation by surface temp erature, but not entirely. More sophisticated application of physics to a sp ectrum allows one to determine separately the temp erature (which lines, and which ionization states are present) the surface gravity (at a given temp erature, a line will b e more pressure- broadened if the surface gravity is larger) chemistry or elemental abundances (relative strength of lines from dif- ferent elements) 5.1.1 Elemental abundances, p opulations I and I I The last classi cation ab ove, elemental abundances, reveals things not only ab out stars in isolation, but also things ab out the history of the Universe, i.e. 113 cosmology.To refresh your memory ab out the elements, here is the Perio dic Table. The numb ers shown on the p erio dic table are the atomic numb ers or num- b ers of protons, Z , in the nucleus. This of course determines the number of electrons and hence the chemistry. For nuclear pro cesses, however, we also care ab out the total number of nucleons (protons plus neutrons), A,in the nucleus. For all the elements from He (not H!) through S (sulfur), the relationship is A 2Z within one AMU. That is, there are close to equal numb ers of protons and neutrons. Heavier elements get slightly richer in neutrons so A b ecomes somewhat larger than 2Z (but not much). Iron, for example has Z =26;A 56. Studying their sp ectra, it is found that there are two distinct classes of stars, called Population I and Population I I (usually read as \Pop One" and \Pop Two"). Pop I stars are young stars, like the Sun, created by 114 ongoing star-formation pro cesses within our Galaxy (or in other galaxies). Pop I I stars are old stars and are thought to fossilize the initial epoch of star formation after the Big Bang. When this classi cation was rst devised, the connection with age was not immediatelyobvious, so the numb ering scheme (\I" and \I I") is p erhaps illogical. A go o d way to rememb er it is to recall that a I I-year-old child is older than a I-year-old child. The observational distinction b etween Pop I and Pop I I is that Pop I stars (e.g., the Sun) are relatively rich in \heavier" elements. For astronomers, \heavier" means anything b eyond H and He in the p erio dic table, that is, Li, Be, B, C, N, O, etc. Chemists would consider these to b e light elements! Another astronomer's habit of whichyou should b e aware is that the \heavy" (though not really heavy!) elements are often referred to lo osely as \metals", even though they're not. So, to astronomers, the whole p erio dic table is reduced to H, He, and \metals". We know quite a lot ab out the elemental abundance of Pop I stars, b e- cause wehave one right in our neighb orho o d (the Sun), and also b ecause, except for H and He which escap ed into space, the Earth itself has in most resp ects the exact elemental abundances of a typical Pop I star. Here is a table, and also graph (note logarithmic scale!) of these \Solar System abundances". 115 Pop I Abundances Mass Atomic Number (main isotop e) Relative Number Mass Fraction H 1 1 1 0.77 2 He 2 4 7 10 0.21 9 > 4 3 > C 6 12 4 10 4 10 > > > 5 3 > > N 7 14 9 10 1 10 > > > 4 3 > = O 8 16 7 10 9 10 \metals" 4 3 Ne 10 20 1 10 1 10 > total 0.02. > > 5 4 > Mg 12 24 4 10 8 10 > > > > 5 4 > Si 14 28 4 10 8 10 > > ; 5 3 Fe 26 56 3 10 1 10 H 1 He 10-1 10-2 10-3 -4 C O Ne Mg Si S Fe 10 N -5 Al Ca Ni 10 Na Ar -6 Cr 10 P Cl K Ti Mn -7 Co Zn 10 F V Cu -8 10 Sc -9 10 B 10-10 Li Be 10-11 -12 10 1234567 8 9 101112131415161718192021222324252627282930 Atomic Number Hydrogen and Helium are primordial to the big bang (in fact, the Helium is pro duced during the rst few minutes of the big bang). All the elements from Carb on on are pro duced solely in stars. The combined mass fraction of \metals" (Carb on and heavier) is usually denoted Z ,soPop I stars have typically Z 0:02. (Don't confuse this use of the letter Z with its other use, denoting atomic numb er. They are unrelated.) Note the very low abundances of Li, Be, B. These are hardly pro duced in the big bang at all, and they are in fact destroyed by stars { they get \co oked" into heavier elements. Also note that the even elements tend to b e more abundant(by a factor of order 10) than their o dd neighb ors. This is 116 b ecause they get \built" eciently out of -particles (Helium nucleii) that contain 2 protons + 2 neutrons. So, Pop I stars are made of material that has already been processed through an earlier generation of stars, either Pop I I stars or else older Pop I stars (whose creation and destruction is a continuous pro cess). In fact, many astrophysicists study what amounts to the \ecology" of stars, that is, pro cesses by which they form, evolve, are disrupted, and return their now- pro cessed material to the interstellar medium for formation in a subsequent generation of stars. Since Pop I I stars were (as far as we know) the original generation of stars, you would exp ect them to havemuchlower metal abundances. Indeed that is true. Typically they mighthave Z 0:002, ab out 1=10 of the Solar abundances. Some haveeven smaller Z values. In our Galaxy,Pop I stars are lo cated in the disk, while Pop I I stars are in the so-called bulge and halo. Thus, we know that these parts of the Galaxy, b eing p opulated with primordial stars, are older. 5.1.2 Nuclear reactions Stars are p owered bynuclear reactions that transmute (we sometimes lo osely say \burn") lighter elements into heavier ones. Main-sequence stars are all powered by the simplest p ossible transmutation, namely of four Hydrogen nucleii (protons) into one Helium nucleus. The reason that burning H to He pro duces energy is of course the fact that the He nucleus weighs slightly less than the 4 H's. In atomic mass units (AMUs): 117 4 M =41:008 ! 1 M =4:0027 : H He That is M 4:032 4:0027 = =0:007 : 4M 4:032 H 2 So, 0:7% of the mass of each proton is converted to energy (E = mc ). In practice, b ecause of the various rules governing nuclear reactions and their probabilities, the reaction is not simply 1 1 1 1 4 H+ H+ H+ H ! He [not!] (The leading sup erscripts are used to indicate mass numb er.) For one thing, the ab ove do es not conservecharge! Instead, the reaction pro ceeds by a series of 2-b o dy interactions. In lower mass stars (< 1:5M ) the so-called p p cycle dominates: 1 1 2 + H+ H ! D + e + 2 1 3 D+ H ! He + 3 3 4 1 1 He + He ! He + H+ H Higher mass stars (> 1:5M ) \burn" via the CNO cycle: 12 1 13 C+ H ! N+ decays 13 13 + N ! C+e + 13 1 14 C+ H ! N+ 14 1 15 N+ H ! O+ decays 15 15 + O ! N+e + 15 1 12 4 N+ H ! C+ He : 118 12 Notice that the Cisacatalyst in this case: it comes into the cycle at the top and is regenerated at the b ottom. The net reaction involves only the particles shown ab ove in b oldface, namely 1 4 + 4 H ! He + 2e +2: (The 's are just energy coming o .) The key fact ab out nuclear reactions is that they are extremely temp er- ature sensitive. That is, ab ove a certain threshold, a small increase in tem- p erature makes a huge increase in reaction rate. This fact makes most stars thermally very constant: across a wide range of stellar masses, the central temp eratures of stars (where the nuclear burning takes place) are b etween 7 7 1 10 K and 2 10 K. That is, this small temp erature range translates into the entire range of luminosity needed to \hold up" against gravity b oth massive stars (high luminosity) and light stars (low luminosity). A good empirical approximation for the central temp erature of main-sequence stars is ! 1=3 M 7 T 1:5 10 K : c M 5.2 Stellar structure 5.2.1 Order-of-magnitude stellar structure The virial theorem, whichwe previously derived for a collection of gravitating b o dies, applies equally well to any combination of inverse-square-law forces between b o dies.